I have a good friend who tutors kids in math. He feels that even within mathematics, there is ''room for interpretation,'' as he puts it. It got me to thinking - could hermeneutics offer an approach to describing the nature of developing and understanding mathematical equations and processes? I've always viewed math as a very cut and dry type of subject, with no room for individual interpretation. My friend tried to explain this to me a bit further in that when he is tutoring his students in math, he isn't only concerned with helping them to arrive at the correct answers, but he wants them to explore different ways of arriving at the correct answers. Even if the answer is incorrect, he said that the path as to how they arrived at the wrong answer will help them to find the right answer, and that is often left up to interpretation, because no two students may process the equation in the same way.

It is in the ''how'' one arrives at the answer, that is just as fulfilling he claims, as actually arriving at the answer. So, I suppose that is where hermeneutics would come into the equation, no pun.

He added that maybe my disdain of math is because I've never had a teacher who taught me more than how to memorize a procedure or path, as to how to arrive at the ''right'' answer. Straight up memorization is painfully boring, if you ask me.

So, what is your personal viewpoint on this? Do you believe that there is a place for interpretation (hermeneutics) in math?

I have a good friend who tutors kids in math. He feels that even within mathematics, there is ''room for interpretation,'' as he puts it. It got me to thinking - could hermeneutics offer an approach to describing the nature of developing and understanding mathematical equations and processes? I've always viewed math as a very cut and dry type of subject, with no room for individual interpretation. My friend tried to explain this to me a bit further in that when he is tutoring his students in math, he isn't only concerned with helping them to arrive at the correct answers, but he wants them to explore different ways of arriving at the correct answers. Even if the answer is incorrect, he said that the path as to how they arrived at the wrong answer will help them to find the right answer, and that is often left up to interpretation, because no two students may process the equation in the same way.

It is in the ''how'' one arrives at the answer, that is just as fulfilling he claims, as actually arriving at the answer. So, I suppose that is where hermeneutics would come into the equation, no pun.

He added that maybe my disdain of math is because I've never had a teacher who taught me more than how to memorize a procedure or path, as to how to arrive at the ''right'' answer. Straight up memorization is painfully boring, if you ask me.

So, what is your personal viewpoint on this? Do you believe that there is a place for interpretation (hermeneutics) in math?

*Note - mods, please remove this thread from the math sub forum above, it is better suited in this section.

There are often multiple ways to arrive at the same solution in math.
Simplistically, 24x9 can be done mentally as 20x9+4*9 or as 24x10-24.

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That's more along the lines of procedural knowledge, just following a different set of procedures or memorization paths to arrive at the same answer.

But I don't see what this hermeneutics has to do with it.

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Your above example isn't an example of it. Instead, [how I understand it anyway], is that hermeneutics when applied in math, can break people away from the traditional methodical memorization approaches, that are having students chant in unison, the steps to arriving at an answer. Instead, students should be discussing how they process a math problem in order to arrive at the answer, because it's in their individual creative processes at arriving at correct answers, that will provide them with a deeper understanding of and respect for math, as a whole. And perhaps an enjoyment for math, as well. I think that's the idea.

The formalism of mathematics and logic are strings of symbols linked by syntactic operators. In order for a string of symbols to acquire a meaning, in theoretical physics or whatever it is, the symbols have to be interpreted. The symbols have to stand for something, in other words. The truth-values of logical formulae will be dependent on their interpretations.

There's a whole sub-field of relatively advanced logic that addresses these interpretations and their various properties, called formal semantics.

That's not exactly what hermeneutics means in Biblical and literary interpretation, and in areas like interpreting historical events where these methods have been extended. But it's probably the closest mathematical analogue that I'm aware of.

I have a good friend who tutors kids in math. He feels that even within mathematics, there is ''room for interpretation,'' as he puts it. It got me to thinking - could hermeneutics offer an approach to describing the nature of developing and understanding mathematical equations and processes? I've always viewed math as a very cut and dry type of subject, with no room for individual interpretation. My friend tried to explain this to me a bit further in that when he is tutoring his students in math, he isn't only concerned with helping them to arrive at the correct answers, but he wants them to explore different ways of arriving at the correct answers.

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I don't think that what is being described there is hermeneutics so much as it's heuristics. A heuristic is an approach that facilitates problem solving, learning or discovery. It needn't be a well-defined method or an algorithm, it might be more intuitive and creative than that. (Rules-of-thumb, analogies, educated guesses...) Heuristics are of great interest in psychology, artificial intelligence and even in the philosophy of science (hypothesis formation and originality in laboratory technique). To some extent, intelligence is a function of one's ability to recognize, create and exploit heuristics.

I rather agree with Seattle. Of course it is true that one can discuss the different ways in which people perceive, or tackle, a mathematical object, equation or problem. For example, I tend to use graphical images, seeing Sin x as a wave, Cos x as a wave 90deg ahead of it, and so on. That is one representation - or "interpretation", if you like - of Sin x and Cos x. Others may see these as power series, others again in terms of the ratios of the sides of right-angled triangles. These are all different facets of the same mathematical "thing".

But, like Seattle, I do not think any purpose is served by replacing "interpretation" by "hermeneutics". "Hermeneutics is a word that belongs to religious scholarship and philosophy. I fear that stretching its use to include mathematics would be deeply unhelpful. It would create a highly misleading impression of vagueness or relativism in the subject, whereas mathematics is famously punctilious about definitions and exactness. This, I feel sure, would lead to unfortunate side-effects, surrounding maths with an aura of quasi-mystical obscurantism and, eventually, bullshit (Deepak Chopra et al).

Math has many straightforward operations, like addition and subtraction. Math also uses numerical methods to arrive at solutions. The numerical methods did not appear as naturally, but rather were invented, out of necessity, to help deal with problems that can't be solved in a direct way with standard math. This latter aspect of math is hermeneutic, since there are often several options available, just as there are several ways to interpret a test.

None can come to a final answer, but each can approach the answer to high level of certainty. These became popular with the advent of computers, since they are not easy to solve without the aid of computers.

It would be like approximating 1 +1 =2, by breaking down 1 into an infinite number of dash increments, which we will them summate. With the advent of high speed computers, we can get the answer 2 in a short time, so it looks as valid as normal addition. Or we can approximate 1 as a series of dots, of increasingly small size, with each iteration summation. This too will get us to 2, in a short time, if we use a fast computer.

I think hermeneutics is generally just a term that is used for interpreting the Bible. It's not really used for interpreting anything else is it?

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Hermeneutics in the broad sense is applicable to anything that expresses intention or has meaning. It applies to interpretation of written texts of all sorts (not just religious), to understanding speech, and to interpretation of things like archaeological remains and understanding human behavior more generally (a ritual, a parade, a political rally...). In other words, it applies to any cultural artifact in which mind and psychology went into its creation.

Unlike the sound of the wind or a crack of thunder, texts and speech typically mean something. They aren't just successions of noises or meaningless symbols. They express a purpose, an intention. They were typically meant to communicate something to other people. They arose in a particular context and probably need to be understood in that context. How were the words used understood when they were written or uttered? Who was the author of the text and why might he or she have written it? Who might it have been intended for? Who might have actually read it? What was the intellectual context at the time? How might readers of the time have understood it given that context?

In an even broader sense, understanding space-aliens will be a hermeneutic problem. There's a new science fiction movie coming out in the next couple of months whose protagonist is a linguist assigned to a first contact team charged with trying to communicate with aliens that suddenly show up on Earth. It develops that the aliens' psychology is as radically different from humans' as their anatomy. So it can probably be said that any sentient being that behaves with intelligence and purpose will present interpretive problems when we try to understand what they are doing.

The intuitive understanding of human behavior (including people's attempts at communication) has been around as long as there have been human beings. More scholarly forms of textual interpretation have existed since ancient times, in both ancient Greece and in India. But the idea of hermeneutics as a distinct discipline seems to have arisen in the 19th century, when people were criticizing the mind-body distinction and arguing that human behavior and cultural products are physical phenomena that can be and should be understood by the same means that science understands the rest of the physical world. That created a counter-reaction among the interpretivists, led by a German philosopher named Dilthey who argued that human actions are categorically distinct from other physical events.

Hermeneutics in the broad sense is applicable to anything that expresses intention or has meaning. It applies to interpretation of written texts of all sorts (not just religious), to understanding speech, and to interpretation of things like archaeological remains and understanding human behavior more generally (a ritual, a parade, a political rally...). In other words, it applies to any cultural artifact in which mind and psychology went into its creation.

Unlike the sound of the wind or a crack of thunder, texts and speech typically mean something. They aren't just successions of noises or meaningless symbols. They express a purpose, an intention. They were typically meant to communicate something to other people. They arose in a particular context and probably need to be understood in that context. How were the words used understood when they were written or uttered? Who was the author of the text and why might he or she have written it? Who might it have been intended for? Who might have actually read it? What was the intellectual context at the time? How might readers of the time have understood it given that context?

In an even broader sense, understanding space-aliens will be a hermeneutic problem. There's a new science fiction movie coming out in the next couple of months whose protagonist is a linguist assigned to a first contact team charged with trying to communicate with aliens that suddenly show up on Earth. It develops that the aliens' psychology is as radically different from humans' as their anatomy. So it can probably be said that any sentient being that behaves with intelligence and purpose will present interpretive problems when we try to understand what they are doing.

The intuitive understanding of human behavior (including people's attempts at communication) has been around as long as there have been human beings. More scholarly forms of textual interpretation have existed since ancient times, in both ancient Greece and in India. But the idea of hermeneutics as a distinct discipline seems to have arisen in the 19th century, when people were criticizing the mind-body distinction and arguing that human behavior and cultural products are physical phenomena that can be and should be understood by the same means that science understands the rest of the physical world. That created a counter-reaction among the interpretivists, led by a German philosopher named Dilthey who argued that human actions are categorically distinct from other physical events.

Even working with this broader understanding of the term I have considerable difficulty in the value of applying it to mathematics.

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I do too. See my earlier post, #6 in this thread. In post #11 I was responding to Seattle's question.

The process in which the mathematical symbolism acquires its interpretation, is formal semantics. That's not exactly what Dilthey had in mind, obviously.

I'm not sure if Wegs meant that more humanistic kind of 'hermeneutics' either. What her math tutor friend was trying to do (to teach math problem solving) sounded more like 'heuristics' to me.

Do you think it has applicability and if so can you give an example that leads to insight?

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The more linguistic kind of interpretation might arise at second hand when we try to interpret something like 'F = MA' as 'force equals mass times acceleration'. We are faced with questions about what the words in the resulting interpretation mean.

One of the problems interpreting Aristotle's physics is that he used the word translated as 'motion' in a way that's unfamiliar to moderns, to mean 'change' instead of geometrical translation from place to place. A qualitative change such as a change in intensity was a 'motion' to Aristotle. It's always going to be hard to translate words one-to-one from a language like ancient Greek or Sanskrit into modern English. And more metaphysical interpretive questions are still going to arise even with the modern words. What does 'force' mean, exactly? How is it used in modern physics and in daily life?

Mathematics is dependent on logic and on the idea of logical implication and the meaning of all of the logical connectives. The behavior of the physical world certainly seems to exemplify and illustrate logical and mathematical relationships, but that isn't the same thing as consciously understanding them as human beings try to do. As you like to say, our scientific formulations are a model of physical reality, a model constructed by human beings, in a particular cultural context (in the light of the history of science and mathematics, for one thing). So some of the hermeneutic ideas would seem to me to apply to the language of science, probably including the formal language of mathematics, when it's considered as a human cultural product.

There was a remarkable paper about hermeneutics in Quantum Gravity, by Sokal:

Transgressing the Boundaries: Towards a Transformative Hermeneutics of Quantum Gravity

and it contains some very interesting points about mathematics too. My favored quote:

Just as liberal feminists are frequently content with a minimal agenda of legal and social equality for women and ``pro-choice'', so liberal (and even some socialist) mathematicians are often content to work within the hegemonic Zermelo-Fraenkel framework (which, reflecting its nineteenth-century liberal origins, already incorporates the axiom of equality) supplemented only by the axiom of choice. But this framework is grossly insufficient for a liberatory mathematics, as was proven long ago by Cohen (1966).

I do too. See my earlier post, #6 in this thread. In post #11 I was responding to Seattle's question.

The process in which the mathematical symbolism acquires its interpretation, is formal semantics. That's not exactly what Dilthey had in mind, obviously.

I'm not sure if Wegs meant that more humanistic kind of 'hermeneutics' either. What her math tutor friend was trying to do (to teach math problem solving) sounded more like 'heuristics' to me.

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That could very well be a much more suitable term for what I'm trying to explain, Yazata. What my friend was getting at is that often when it comes to math, students are taught a set of rules and procedural/memorization type problem solving skill sets. I look back to high school for example, where my teachers were terrible at allowing students to find their own way of arriving at the correct answers. What does it matter how I figure a math problem out, as long as the answer in the end is correct? And if it isn't correct, maybe then I learn why some procedural paths are better than others, in solving for the correct answers. I remember one of my high school Algebra teachers who would mark your answer wrong on an exam, even if it was the correct answer, because he would collect the paper you used to formulate the answer and if he didn't approve of your methodology, he marked it wrong. What kind of a human being does this!? lol I can laugh about it now, but back then, I would go to him after class and say ''this is unfair! I got the answer right!'' But, he didn't want to hear it.

The more linguistic kind of interpretation might arise at second hand when we try to interpret something like 'F = MA' as 'force equals mass times acceleration'. We are faced with questions about what the words in the resulting interpretation mean.

One of the problems interpreting Aristotle's physics is that he used the word translated as 'motion' in a way that's unfamiliar to moderns, to mean 'change' instead of geometrical translation from place to place. A qualitative change such as a change in intensity was a 'motion' to Aristotle. It's always going to be hard to translate words one-to-one from a language like ancient Greek or Sanskrit into modern English. And more metaphysical interpretive questions are still going to arise even with the modern words. What does 'force' mean, exactly? How is it used in modern physics and in daily life?

Mathematics is dependent on logic and on the idea of logical implication and the meaning of all of the logical connectives. The behavior of the physical world certainly seems to exemplify and illustrate logical and mathematical relationships, but that isn't the same thing as consciously understanding them as human beings try to do. As you like to say, our scientific formulations are a model of physical reality, a model constructed by human beings, in a particular cultural context (in the light of the history of science and mathematics, for one thing). So some of the hermeneutic ideas would seem to me to apply to the language of science, probably including the formal language of mathematics, when it's considered as a human cultural product.

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Actually, I think that the term serves math well, in the sense that it can serve math TEACHERS well. However, the challenge with teachers are their classroom sizes in public schools, so to allow for individuality and the ability to create one's own path to solving for the correct answers in math, would be difficult because of the time constraints teachers have to drive a point home. This is why tutors can be so helpful though, for the first time my math tutor friend said, there are students he's dealing with who are enjoying math because they are using their own unique problem solving skills to map out a process of solving math equations. Of course, they have to get some basic memorization down with certain rules, but overall, he is letting them plot their own course. I think it's really cool.

I rather agree with Seattle. Of course it is true that one can discuss the different ways in which people perceive, or tackle, a mathematical object, equation or problem. For example, I tend to use graphical images, seeing Sin x as a wave, Cos x as a wave 90deg ahead of it, and so on. That is one representation - or "interpretation", if you like - of Sin x and Cos x. Others may see these as power series, others again in terms of the ratios of the sides of right-angled triangles. These are all different facets of the same mathematical "thing".

But, like Seattle, I do not think any purpose is served by replacing "interpretation" by "hermeneutics". "Hermeneutics is a word that belongs to religious scholarship and philosophy. I fear that stretching its use to include mathematics would be deeply unhelpful. It would create a highly misleading impression of vagueness or relativism in the subject, whereas mathematics is famously punctilious about definitions and exactness. This, I feel sure, would lead to unfortunate side-effects, surrounding maths with an aura of quasi-mystical obscurantism and, eventually, bullshit (Deepak Chopra et al).

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In the definition above, the term's meaning has been broadened to mean interpretation, in a more general sense.

First encounter with the word..hermeneutics. (Thank God auto completion did the trick).

The question is still not clear to me but solutions of mathematical equations are open to interpretation.

For example IMO Black Hole is a bad interpretation of the solutions of Einstein Field Equations. Similarly time travel also is a bad interpretation of maths.

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Well, I think that it doesn't quite mean that we can have any and all interpretations of objective truths, thus becoming subjective. When people apply interpretation to math equation problem solving, the end result still must be correct or it's not interpretation, it's just wrong. lol

That could very well be a much more suitable term for what I'm trying to explain, Yazata. What my friend was getting at is that often when it comes to math, students are taught a set of rules and procedural/memorization type problem solving skill sets. I look back to high school for example, where my teachers were terrible at allowing students to find their own way of arriving at the correct answers. What does it matter how I figure a math problem out, as long as the answer in the end is correct? And if it isn't correct, maybe then I learn why some procedural paths are better than others, in solving for the correct answers. I remember one of my high school Algebra teachers who would mark your answer wrong on an exam, even if it was the correct answer, because he would collect the paper you used to formulate the answer and if he didn't approve of your methodology, he marked it wrong. What kind of a human being does this!? lol I can laugh about it now, but back then, I would go to him after class and say ''this is unfair! I got the answer right!'' But, he didn't want to hear it.

Actually, I think that the term serves math well, in the sense that it can serve math TEACHERS well. However, the challenge with teachers are their classroom sizes in public schools, so to allow for individuality and the ability to create one's own path to solving for the correct answers in math, would be difficult because of the time constraints teachers have to drive a point home. This is why tutors can be so helpful though, for the first time my math tutor friend said, there are students he's dealing with who are enjoying math because they are using their own unique problem solving skills to map out a process of solving math equations. Of course, they have to get some basic memorization down with certain rules, but overall, he is letting them plot their own course. I think it's really cool.